L(s) = 1 | + (2.36 − 1.36i)2-s + (−1.43 + 0.972i)3-s + (2.72 − 4.71i)4-s + 5-s + (−2.05 + 4.25i)6-s + (1.49 + 2.18i)7-s − 9.39i·8-s + (1.10 − 2.78i)9-s + (2.36 − 1.36i)10-s + 0.284i·11-s + (0.686 + 9.40i)12-s + (−2.89 + 1.67i)13-s + (6.50 + 3.12i)14-s + (−1.43 + 0.972i)15-s + (−7.36 − 12.7i)16-s + (−2.65 − 4.60i)17-s + ⋯ |
L(s) = 1 | + (1.67 − 0.964i)2-s + (−0.827 + 0.561i)3-s + (1.36 − 2.35i)4-s + 0.447·5-s + (−0.840 + 1.73i)6-s + (0.564 + 0.825i)7-s − 3.32i·8-s + (0.368 − 0.929i)9-s + (0.747 − 0.431i)10-s + 0.0856i·11-s + (0.198 + 2.71i)12-s + (−0.803 + 0.464i)13-s + (1.73 + 0.835i)14-s + (−0.369 + 0.251i)15-s + (−1.84 − 3.19i)16-s + (−0.644 − 1.11i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.429 + 0.903i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.429 + 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.29477 - 1.45004i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.29477 - 1.45004i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.43 - 0.972i)T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (-1.49 - 2.18i)T \) |
good | 2 | \( 1 + (-2.36 + 1.36i)T + (1 - 1.73i)T^{2} \) |
| 11 | \( 1 - 0.284iT - 11T^{2} \) |
| 13 | \( 1 + (2.89 - 1.67i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.65 + 4.60i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.743 - 0.429i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 7.91iT - 23T^{2} \) |
| 29 | \( 1 + (1.98 + 1.14i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.992 - 0.572i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.53 - 7.84i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.15 - 5.46i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.223 + 0.387i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3.00 + 5.21i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.483 + 0.279i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.59 + 7.96i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6.96 - 4.02i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.802 - 1.39i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 8.30iT - 71T^{2} \) |
| 73 | \( 1 + (-3.55 + 2.05i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.29 + 3.97i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.29 + 3.97i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-5.97 + 10.3i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (3.69 + 2.13i)T + (48.5 + 84.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.70095659205636394150160504484, −11.05988459004767788120024874778, −9.912096628721747823254241297437, −9.349195624178982263639828871181, −7.04013670688488181187009149981, −5.98643975028228683164058245542, −5.13819697479564786421303705061, −4.61779014889822603370077987497, −3.15732067414506575423630720891, −1.80446712173957139655593614522,
2.29229727531155416597606695042, 4.10220099444776340233116746294, 4.95791438282671847171051991407, 5.85334076556701079411364419403, 6.74553714326905446119326158204, 7.46009621634910682016724409294, 8.390922674079897521334310043264, 10.52975137959220985719363807106, 11.14319169562857404214487801616, 12.38099152853238987442451405315